Theorem nelpri

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015)

	Ref	Expression
Hypotheses	nelpri.1	⊢ 𝐴 ≠ 𝐵
	nelpri.2	⊢ 𝐴 ≠ 𝐶
Assertion	nelpri	⊢ ¬ 𝐴 ∈ { 𝐵 , 𝐶 }

Proof

Step	Hyp	Ref	Expression
1		nelpri.1	⊢ 𝐴 ≠ 𝐵
2		nelpri.2	⊢ 𝐴 ≠ 𝐶
3		neanior	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) )
4		elpri	⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 } → ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) )
5	4	con3i	⊢ ( ¬ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) → ¬ 𝐴 ∈ { 𝐵 , 𝐶 } )
6	3 5	sylbi	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) → ¬ 𝐴 ∈ { 𝐵 , 𝐶 } )
7	1 2 6	mp2an	⊢ ¬ 𝐴 ∈ { 𝐵 , 𝐶 }